Transcript QCD - MSU
QCD Hsiang-nan Li Academia Sinica, Taipei Presented at AEPSHEP Oct. 18-22, 2012 Titles of lectures • • • • Lecture I: Factorization theorem Lecture II: Evolution and resummation Lecture III: PQCD for Jet physics Lecture IV: Hadronic heavy-quark decays References • Partons, Factorization and Resummation, TASI95, G. Sterman, hep-ph/9606312 • Jet Physics at the Tevatron , A. Bhatti and D. Lincoln, arXiv:1002.1708 • QCD aspects of exclusive B meson decays, H.-n. Li, Prog.Part.Nucl.Phys.51 (2003) 85, hep-ph/0303116 Lecture I Factorization theorem Hsiang-nan Li Oct. 18, 2012 Outlines • • • • • QCD Lagrangian and Feynman rules Infrared divergence and safety DIS and collinear factorization Application of factorization theorem kT factorization QCD Lagrangian See Luis Alvarez-Gaume’s lectures Lagrangian • SU(3) QCD Lagrangian • Covariant derivative, gluon field tensor • Color matrices and structure constants Gauge-fixing • Add gauge-fixing term to remove spurious degrees of freedom • Ghost field from Jacobian of variable change, as fixing gauge Feynman rules Feynman rules Asymptotic freedom • QCD confinement at low energy, hadronic bound states: pion, proton,… • Manifested by infrared divergences in perturbative calculation of bound-state properties • Asymptotic freedom at high energy leads to small coupling constant • Perturbative QCD for high-energy processes Infrared divergence and safty Vertex correction • Start from vertex correction as an example • Inclusion of counterterm is understood Light-cone coordinates • Analysis of infrared divergences simplified l (l , l , lT ) l l l 2 0 3 • As particle moves along light cone, only one large component is involved Leading regions • • • • Collinear region Soft region Infrared gluon Hard region l (l , l , lT ) ~ ( E , 2 E , ) l ~ ( , , ) l 2 ~ 2 l ~ (E, E, E ) • They all generate log divergences d 4l d 4 d 4E l 4 ~ 4 ~ E 4 ~ log Contour integration • In terms of light-cone coordinates, vertex correction is written as • Study pole structures, since IR divergence comes from vanishing denominator Pinched singularity • Contour integration over lNon-pinch • collinear region 1 3 • Soft region 1 3 Double IR poles • Contour integration over l- gives e+e- annihilation • calculate e+e- annihilation • cross section = |amplitude|2 • Born level final-state cut fermion charge momentum transfer squared Real corrections • Radiative corrections reveal two types of infrared divergences from on-shell gluons • Collinear divergence: l parallel P1, P2 • Soft divergence: l approaches zero overlap of collinear and soft divergences Virtual corrections • Double infrared pole also appears in virtual corrections with a minus sign overlap of collinear and soft divergences Infrared safety • Infrared divergences cancel between real and virtual corrections • Imaginary part of off-shell photon self-energy corrections • Total cross section (physical quantity) of e+e- -> X is infrared safe i 2 Im 2 (p ) p i propagator on-shell final state KLN theorem • Kinoshita-Lee-Neuberger theorem: IR cancellation occurs as integrating over all phase space of final states • Naïve perturbation applies • Used to determine the coupling constant DIS and collinear factorization Deep inelastic scattering • Electron-proton DIS l(k)+N(p) -> l(k’)+X • Large momentum transfer -q2=(k-k’)2=Q2 • Calculation of cross section suffers IR divergence --- nonperturbative dynamics in the proton • Factor out nonpert part from DIS, and leave it to other methods? Structure functions for DIS • Standard example for factorization theorem LO amplitude NLO diagrams NLO total cross section LO term plus function infrared divergence IR divergence is physical! Hard dynamics q g q t=-infty Soft dynamics t=0, when hard scattering occurs • It’s a long-distance phenomenon, related to confinement. • All physical hadronic high-energy processes involve both soft and hard dynamics. Collinear divergence • Integrated over final state kinematics, but not over initial state kinematics. KLN theorem does not apply • Collinear divergence for initial state quark exists. Confinement of initial bound state • Soft divergences cancel between virtual and real diagrams (proton is color singlet) • Subtracted by PDF, evaluated in hard kernel or Wilson coefficient perturbation Assignment of IR divergences Parton distribution function • Assignment at one loop • PDF in terms of hadronic matrix element reproduces IR divergence at each order splitting kernel Wilson links Factorization at diagram level Eikonal approximation P q l k l P q , k k , P P q q 2 2 ( Pq l ) (k l ) P q l k l P q , l l ( Pq l ) 2 (k l )2 P q l k P q 2 2 Pq l (k l ) Pq l P q 2 Pq P q 2 Pq l k l 2 , P P P q q q 0 2 (k l ) k l n P q 2 (k l ) n l k Effective diagrams • Factorization of collinear gluons at leading power leads to Wilson line W(y-,0) necessary for gauge invariance • Collinear gluons also change parton momentum ~ Wilson links loop momentum flows through the hard kernel y0 y- loop momentum does not flow through the hard kernel 0 Factorization in fermion flow • To separate fermion flows for H and for PDF, insert Fierz transformation i j k l • ( )lj 2 ( )lj 2 goes into definition of PDF. Others contribute at higher powers Factorization in color flow • To separate color flows for H and for PDF, insert Fierz transformation i j for color-octet state, namely for three-parton PDF • I lj NC goes into definition of PDF k l Parton model • The proton travels huge space-time, before hit by the virtual photon • As Q2 >>1, hard scattering occurs at point space-time • The quark hit by the virtual photon behaves like a free particle • It decouples from the rest of the proton • Cross section is the incoherent sum of the scattered quark of different momentum Incoherent sum 2 i 2 i holds after collinear factorization Factorization formula • DIS factorized into hard kernel (infrared finite, perturbative) and PDF (nonperturbative) F ( x ) f (d ) H f ( x ) f N ( ) 1 x • Universal PDF describes probability of parton f carrying momentum fraction in nucleon N • PDF computed by nonpert methods, or extracted from data k ( P ,0,0T ) Expansion on light cone • Operator product expansion (OPE): expansion in small distance y • Infrared safe ee X iCi ( y)Oi (0) 0 y 2 y • Factorization theorem: expansion in • Example: Deeply inelastic scattering (DIS) • Collinear divergence in longitudinal direction exists (particle travels) finite y Factorization scheme • Definition of an IR regulator is arbitrary, like an UV regulator: (1) ~1/IR+finite part • Different inite parts shift between and H correspond to different factorization schemes • Extraction of a PDF depends not only on powers and orders, but on schemes. • Must stick to the same scheme. The dependence of predictions on factorization schemes would be minimized. 2 Extraction of PDF • Fit the factorization formula F=HDIS f/N to data. Extract f/N for f=u, d, g(luon), sea CTEQ-TEA PDF NNLO: solid color NLO: dashed NLO, NNLO means Accuracy of H Nadolsky et al. 1206.3321 PDF with RG see Lecture 2 Application of factorization theorem Hard kernel • PDF is infrared divergent, if evaluated in perturbation confinement • Quark diagram is also IR divergent. • Difference between the quark diagram and PDF gives the hard kernel HDIS HDIS= _ Drell-Yan process • Derive factorization theorem for Drell-Yan process N(p1)+N(p2)->+-(q)+X p1 Same PDF p2 f/N 1 p 1 X * 2 p 2 f/N + - X Hard kernel for DY • Compute the hard kernel HDY • IR divergences in quark diagram and in PDF must cancel. Otherwise, factorization theorem fails HDY = _ Same as in DIS Prediction for DY • Use DY=f1/N HDY f2/N to make predictions for DY process f1/N DY= HDY f2/N Predictive power • Before adopting PDFs, make sure at which power and order, and in what scheme they are defined Nadolsky et al. 1206.3321 kT factorization Collinear factorization • Factorization of many processes investigated up to higher twists • Hard kernels calculated to higher orders • Parton distribution function (PDF) evolution from low to high scale derived (DGLAP equation) • PDF database constructed (CTEQ) • Logs from extreme kinematics resummed • Soft, jet, fragmentation functions all studied Why kT factorization • kT factorization has been developed for small x physics for some time • As Bjorken variable xB=-q2/(2p.q) is small, parton momentum fraction x > xB can reach xp ~ kT . kT is not negligible. • xp ~ kT also possible in low qT spectra, like direct photon and jet production • In exclusive processes, x runs from 0 to 1. The end-point region is unavoidable • But many aspects of kT factorization not yet investigated in detail Condition for kT factorization • Collinear and kT factorizations are both fundamental tools in PQCD • x 0 (large fractional momentum exists) is assumed in collinear factorization. • If small x not important, collinear factorization is self-consistent • If small x region is important • x 0 y , expansion in y 2 fails • kT factorization is then more appropriate Parton transverse momentum • Keep parton transverse momentum in H • kT dependence introduced by gluon emission • Need to describe distribution in kT F ( x ) f (d ) d kT H f ( x , kT ) f N ( , kT ) 1 x l , kT lT 2 Eikonal approximation P q l k l P q , k k , P P q q 2 2 ( Pq l ) (k l ) P q l k l P q , l l ( Pq l ) 2 (k l )2 P q l k P q 2 2 Pq l (k l ) P q 2 Pq P q 2 Pq l drop lT in numerator to get Wilson line k l , P q P q 0 2 (k l ) Pq k l n P q 2 (k l ) n l l k Effective diagrams • • • • • Parton momentum k ( P ,0, kT ) Only minus component is neglected kT appears only in denominator 2 Collinear divergences regularized by kT Factorization of collinear gluons at leading power leads to Wilson links W(y-,0) ~ kT Factorization in kT space Universal transverse-momentum-dependent (TMD) PDF f N ( , kT ) describes probability of parton carrying momentum fraction and transverse momentum If neglecting kT in H, integration over kT can be worked out, giving d 2 kT f N ( , kT ) f / N ( ) Summary • Despite of nonperturbative nature of QCD, theoetical framework with predictive power can be developed • It is based on factorization theorem, in which nonperturbative PDF is universal and can be extracted from data, and hard kernel can be calculated pertuebatvely • kT factorization is more complicated than collinear factorization, and has many difficulties